In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case ...In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.展开更多
The Fourier Transform Infrared(FTIR)and Fourier transform Raman(FT-Raman)spectra of 1,3-Diphenyl Propenone were recorded in the regions 4000~400 and 4000~100 cm^-1,respectively,in the solid phase.Molecular electronic ...The Fourier Transform Infrared(FTIR)and Fourier transform Raman(FT-Raman)spectra of 1,3-Diphenyl Propenone were recorded in the regions 4000~400 and 4000~100 cm^-1,respectively,in the solid phase.Molecular electronic energy,geometrical structure,harmonic vibrational spectra was computed at the DFT/6-31G(d,p)and three parameter hybrid functional Lee-Yang-Parr/6-31G(d,p)levels of theory.The vibrational studies were interpreted in terms of potential energy distribution(PED).The results were compared with experimental values with the help of scaling procedures.Most of the modes have wave numbers in the expected range and are in good agreement with computed values and also the molecular properties of Mulliken population analysis have been calculated.Besides,thermodynamic properties were performed.展开更多
Pb_ 1-xLa_xTi_ 1-x/4O_3 nanocrystals with different amounts of lanthanum were prepared by modified sol-gel method and detected by FT-Raman spectra at 288 and 378 K, respectively. The improved synthetic technique used ...Pb_ 1-xLa_xTi_ 1-x/4O_3 nanocrystals with different amounts of lanthanum were prepared by modified sol-gel method and detected by FT-Raman spectra at 288 and 378 K, respectively. The improved synthetic technique used in this paper could obtain nanocrystals with apparently promoted synthetic speed from a few days to several hours (4 times faster of hydrolytic process than that of traditional sol-gel method and reduced preparation period of gel from 24 h to several hours) at relatively lower temperature by hydrolysis of sol to gel through adding methyl alcohol/water solution into sol under vigorous stirring and annealed the fresh gel directly. The effects of lanthanum′s amount, annealed temperature and nanocrystals′ grain size of Pb_ 0.9La_ 0.1Ti_ 0.975O_3 nanocrystals on frequency, shape and width of the phonon peak were also discussed in detail.展开更多
Single crystals of the 4,4’-diamoniumdiphenylmethan sulfate hydrate, denoted DDPS, were grown by slow evaporation solution technique at room temperature. The compound was characterized by single crystal X-ray diffrac...Single crystals of the 4,4’-diamoniumdiphenylmethan sulfate hydrate, denoted DDPS, were grown by slow evaporation solution technique at room temperature. The compound was characterized by single crystal X-ray diffraction, IR and thermal analysis (TG-DTA). It crystallizes in the monoclinic system (space group C2/c) with the following unit cell dimensions: a = 17.7635(10) ?, b = 9.3796(10) ?, c = 27.5676(10) ?, β = 97.37(2)°, V = 4555.2(6) ?. The structure was solved by the direct method and refined to final R value of 0.0472 for 2108 independent reflections. The anions are hydrogen bonded to each other, forming clusters [HS2O8]3﹣ parallel to the plan (a, b). The water molecules connect these clusters via O-H···O hydrogen bonds. The organic cations are attached to the clusters through N-H···O hydrogen bonds, forming an infinite three-dimensional network.展开更多
Single crystals of the tetraethylammonium dihydrogenmonophosphate bis trihydrogenmonophosphate [CH3CH2]4N+(H2PO4)(H3PO4)2(TEP), were grown by slow evaporation solution technique at room temperature. The compound was c...Single crystals of the tetraethylammonium dihydrogenmonophosphate bis trihydrogenmonophosphate [CH3CH2]4N+(H2PO4)(H3PO4)2(TEP), were grown by slow evaporation solution technique at room temperature. The compound was characterised by IR, Raman, differential thermal analysis (TG-DTA) and single crystal X-ray diffraction. It crystallizes in the monoclinic system (space group P21/c) with the following unit cell dimensions: a = 7.765 (2) , b = 16.531 (4) , c = 14.843 (2) , β = 100.99 (2)o, Z=4 , DX =1.67Mg.m-3, Dm=1.532Mg-3, λ (MoKa) = 0.71073 , u=0.384 mm-1, F(000)=991, T = 20 (2)o. The structure was solved by the direct method and refined to final R value of 0.0342 and Rw=0.107 for 3239 independent [(CH3CH2)4N]+ reflections. The structure consists of infinite parallel two-dimensional planes built of mutually , H3PO4 tetrahedra ?and cations connected by strong and hydrogen bonding. There are no contacts other than van der Waals interactions between the layers.展开更多
In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on...In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].展开更多
文摘In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.
文摘The Fourier Transform Infrared(FTIR)and Fourier transform Raman(FT-Raman)spectra of 1,3-Diphenyl Propenone were recorded in the regions 4000~400 and 4000~100 cm^-1,respectively,in the solid phase.Molecular electronic energy,geometrical structure,harmonic vibrational spectra was computed at the DFT/6-31G(d,p)and three parameter hybrid functional Lee-Yang-Parr/6-31G(d,p)levels of theory.The vibrational studies were interpreted in terms of potential energy distribution(PED).The results were compared with experimental values with the help of scaling procedures.Most of the modes have wave numbers in the expected range and are in good agreement with computed values and also the molecular properties of Mulliken population analysis have been calculated.Besides,thermodynamic properties were performed.
文摘Pb_ 1-xLa_xTi_ 1-x/4O_3 nanocrystals with different amounts of lanthanum were prepared by modified sol-gel method and detected by FT-Raman spectra at 288 and 378 K, respectively. The improved synthetic technique used in this paper could obtain nanocrystals with apparently promoted synthetic speed from a few days to several hours (4 times faster of hydrolytic process than that of traditional sol-gel method and reduced preparation period of gel from 24 h to several hours) at relatively lower temperature by hydrolysis of sol to gel through adding methyl alcohol/water solution into sol under vigorous stirring and annealed the fresh gel directly. The effects of lanthanum′s amount, annealed temperature and nanocrystals′ grain size of Pb_ 0.9La_ 0.1Ti_ 0.975O_3 nanocrystals on frequency, shape and width of the phonon peak were also discussed in detail.
文摘Single crystals of the 4,4’-diamoniumdiphenylmethan sulfate hydrate, denoted DDPS, were grown by slow evaporation solution technique at room temperature. The compound was characterized by single crystal X-ray diffraction, IR and thermal analysis (TG-DTA). It crystallizes in the monoclinic system (space group C2/c) with the following unit cell dimensions: a = 17.7635(10) ?, b = 9.3796(10) ?, c = 27.5676(10) ?, β = 97.37(2)°, V = 4555.2(6) ?. The structure was solved by the direct method and refined to final R value of 0.0472 for 2108 independent reflections. The anions are hydrogen bonded to each other, forming clusters [HS2O8]3﹣ parallel to the plan (a, b). The water molecules connect these clusters via O-H···O hydrogen bonds. The organic cations are attached to the clusters through N-H···O hydrogen bonds, forming an infinite three-dimensional network.
文摘Single crystals of the tetraethylammonium dihydrogenmonophosphate bis trihydrogenmonophosphate [CH3CH2]4N+(H2PO4)(H3PO4)2(TEP), were grown by slow evaporation solution technique at room temperature. The compound was characterised by IR, Raman, differential thermal analysis (TG-DTA) and single crystal X-ray diffraction. It crystallizes in the monoclinic system (space group P21/c) with the following unit cell dimensions: a = 7.765 (2) , b = 16.531 (4) , c = 14.843 (2) , β = 100.99 (2)o, Z=4 , DX =1.67Mg.m-3, Dm=1.532Mg-3, λ (MoKa) = 0.71073 , u=0.384 mm-1, F(000)=991, T = 20 (2)o. The structure was solved by the direct method and refined to final R value of 0.0342 and Rw=0.107 for 3239 independent [(CH3CH2)4N]+ reflections. The structure consists of infinite parallel two-dimensional planes built of mutually , H3PO4 tetrahedra ?and cations connected by strong and hydrogen bonding. There are no contacts other than van der Waals interactions between the layers.
文摘In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].
